Normal states are determined by their facial distances II

Abstract

AbstractLet be a von Neumann algebra with separable predual. For a normal semi‐finite weight on , denote by the von Neumann subalgebra generated by Let be the center of , and be the set of normal semi‐finite weights on . When has no type part (but could have a non‐trivial type part), for every faithful weights with being strictly semi‐finite, if , then there is a positive self‐adjoint operator affiliated with such that . This does not hold for the hyper‐finite type factor. When has no type part, we verify that for strictly semi‐finite weights with and , one has . This is not true for the hyper‐finite type factor. Denote by the subset of consisting of weights with being semi‐finite. When is the direct sum of a semi‐finite algebra and a type algebra, we show that for , if and , then . This fails for any type factor when . Using the above, we establish that when has no type part, the distances of a normal state on to closed faces of the normal state space of uniquely determine this state.